Summary
We propose a conjecture combining the Mordell-Lang conjecture with an important special case of the André-Oort conjecture, and explain how existing results imply evidence for it.
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References
Y. André — G-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989.
—,Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire”, J. Reine Angew. Math. 505 (1998), p. 203–208.
—,Shimura varieties, subvarieties, and CM points”, Six lectures at the University of Hsinchu (Taiwan), August–September 2001 (with an appendix by C.-L. Chai), http: //www.math.umd.edu/~yu/notes.shtml.
F. A. Bogomolov —Points of finite order on abelian varieties”, Izv. Akad. Nauk SSSR Ser. Mat.44 (1980), no. 4, p. 782–804.
E. Bombieri —The Mordell conjecture revisited”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)17 (1990), no. 4, p. 615–640.
L. Clozel, H. Oh and E. Ullmo —Hecke operators and equidistribution of Hecke points”, Invent. Math.144 (2001), no. 2, p. 327–351.
L. Clozel and E. Ullmo —Équidistribution des points de Hecke”, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, p. 193–254.
R. F. Coleman —Ramified torsion points on curves”, Duke Math. J.54 (1987), no. 2, p. 615–640.
P. Deligne —Travaux de Shimura”, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Springer, Berlin, 1971, p. 123–165. Lecture Notes in Math., Vol. 244.
—,Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques”, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, p. 247–289.
P. Deligne, J. S. Milne, A. Ogus and K.-Y. Shih — Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982.
B. Edixhoven and J.-H. Evertse (eds.) — Diophantine approximation and abelian varieties, Lecture Notes in Mathematics, vol. 1566, Springer-Verlag, Berlin, 1993, Introductory lectures, Papers from the conference held in Soesterberg, April 12–16, 1992.
B. Edixhoven —Special points on the product of two modular curves”, Compositio Math.114 (1998), no. 3, p. 315–328.
—,On the André-Oort conjecture for Hilbert modular surfaces”, Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkhäuser, Basel, 2001, p. 133–155.
B. Edixhoven and A. Yafaev —Subvarieties of Shimura varieties”, Ann. of Math. (2)157 (2003), no. 2, p. 621–645.
G. Faltings —Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math.73 (1983), no. 3, p. 349–366, Erratum: 75 no. 2 (1984).
—,Finiteness theorems for abelian varieties over number fields”, Arithmetic geometry (Storrs, Conn., 1984) (G. Cornell and J. H. Silverman, eds.), Springer, New York, 1986, p. 9–27.
—,Diophantine approximation on abelian varieties”, Ann. of Math. (2) 133 (1991), no. 3, p. 549–576.
—,The general case of S. Lang’s conjecture”, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, p. 175–182.
M. Hindry —Autour d’une conjecture de Serge Lang”, Invent. Math.94 (1988), no. 3, p. 575–603.
—,Sur les conjectures de Mordell et Lang (d’après Vojta, Faltings et Bombieri)”, Astérisque (1992), no. 209, p. 11, 39–56, Journées Arithmétiques, 1991 (Geneva).
M. Hindry and J. H. Silverman — Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000.
E. Hrushovski —The Manin-Mumford conjecture and the model theory of difference fields”, Ann. Pure Appl. Logic112 (2001), no. 1, p. 43–115.
S. Lang —Division points on curves”, Ann. Mat. Pura Appl. (4)70 (1965), p. 229–234.
—, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983.
M. McQuillan —Division points on semi-abelian varieties”, Invent. Math.120 (1995), no. 1, p. 143–159.
J. S. Milne —The action of an automorphism of C on a Shimura variety and its special points”, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, p. 239–265.
—,Canonical models of (mixed) Shimura varieties and automorphic vector bundles”, Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, p. 283–414.
B. Moonen —Special points and linearity properties of Shimura varieties”, Thesis, University of Utrecht, 1995.
—,Models of Shimura varieties in mixed characteristics”, Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, p. 267–350.
L. Mordell —On the rational solutions of the indeterminate equations of the third and fourth degrees”, Proc. Camb. Philos. Soc.21 (1922), p. 179–192.
F. Oort —Canonical liftings and dense sets of CM-points”, Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, p. 228–234.
R. Pink —Arithmetical compactification of mixed shimura varieties”, Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 1989, Bonner Mathematische Schriften, 209 (1990).
R. Pink and D. Roessler —On Hrushovski’s proof of the Manin-Mumford conjecture”, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, p. 539–546.
M. Raynaud —Around the Mordell conjecture for function fields and a conjecture of Serge Lang”, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, p. 1–19.
—,Courbes sur une variété abélienne et points de torsion”, Invent. Math. 71 (1983), no. 1, p. 207–233.
—,Sous-variétés d’une variété abélienne et points de torsion”, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, p. 327–352.
K. A. Ribet —Kummer theory on extensions of abelian varieties by tori”, Duke Math. J.46 (1979), no. 4, p. 745–761.
D. Roessler —A note on the Manin-Mumford conjecture”, 2004, preprint.
J.-P. Serre —Lettre à Ken Ribet du 1/1/1981 et du 29/1/1981”, Oeuvres vol. IV, p. 1–20, Springer, 2000.
—,Lettre à Marie-France Vignéras du 10/2/1986”, Oeuvres vol. IV, p. 38–55, Springer, 2000.
—,Résumé des cours de 1985–1986”, Oeuvres vol. IV, p. 33–37, Springer, 2000.
G. Shimura and Y. Taniyama — Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, The Mathematical Society of Japan, Tokyo, 1961.
E. Ullmo —Positivité et discrétion des points algébriques des courbes”, Ann. of Math. (2)147 (1998), no. 1, p. 167–179.
—,Théorie ergodique et géométrie arithmétique”, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, p. 197–206.
P. Vojta —Siegel’s theorem in the compact case”, Ann. of Math. (2)133 (1991), no. 3, p. 509–548.
—,Applications of arithmetic algebraic geometry to Diophantine approximations”, Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Math., vol. 1553, Springer, Berlin, 1993, p. 164–208.
A. Weil —L’arithmétique sur les courbes algébriques”, Acta Math.52 (1928), p. 281–315.
G. Wüstholz —New developments in diophantine and arithmetic algebraic geometry”, Aspects of Mathematics, E6, p. x+311, Friedr. Vieweg & Sohn, Braunschweig, third ed., 1992, Appendix toPapers from the seminar held at the Max-Planck-Institut für Mathematik”, Bonn/Wuppertal, 1983/1984.
A. Yafaev —On a result of Ben Moonen on the moduli space of principally polarised abelian varieties”, to appear in Compositio Math.
—,A conjecture of Yves André”, 2003, math.NT/0302125.
S.-W. Zhang —Equidistribution of small points on abelian varieties”, Ann. of Math. (2)147 (1998), no. 1, p. 159–165.
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Pink, R. (2005). A Combination of the Conjectures of Mordell-Lang and André-Oort. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_11
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DOI: https://doi.org/10.1007/0-8176-4417-2_11
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